Assessing Exploratory Theory Formation Programs

Broadly speaking, machine learning programs are asked to identify a single concept given a set of examples and some background knowledge. Mathematical theory formation programs, such as the AM program, (Davis & Lenat 1982) and the HR program, (Colton, Bundy, & Walsh 1999), are also given a set of examples and some background knowledge. However, they are not asked to find a single concept, but rather to explore the domain and attempt to gain some understanding of it. Because the domain is mathematics, there are a range of ways by which the program can gain an understanding of the domain, including inventing concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. The HR system is able to perform all of these activities in domains such as group theory, where it employs the Otter theorem prover, (McCune 1990), and MACE counterexample finder, (McCune 1994), to prove and disprove theorems.